Multiloop calculations in p-adic string theory and Bruhat-Tits trees

We treat the open p-adic string world sheet as a coset space F=T/Γ, where T is the Bruhat-Tits tree for the p-adic linear group GL(2, Qp) and Γis contained inPGL(2, Qp) is some Schottky group. The boundary of this world sheet corresponds to a p-adic Mumford curve of finite genus. The string dynamics is governed by the local gaussian action on the coset space F. The tachyon amplitudes expressed in terms of p-adic θ-functions are proposed for the Mumford curve of arbitrary genus. We compare them with the corresponding usual archimedean amplitudes. The sum over moduli space of the algebraic curves is conjectured to be expressed in the arithmetic surface terms. We also give the necessary mathematical background including the Mumford approach to p-adic algebraic curves. The connection of the problem of closed p-adic strings with the considered topics is discussed. (orig.)